1.1 Simplicial complexes

Simplicial complexes are the building blocks of algebraic topology, providing a discrete way to model topological spaces. They're made up of simplices - points, lines, triangles, and higher-dimensional analogs - connected in specific ways to create complex structures.

These structures allow us to study the properties of spaces using combinatorial methods. By analyzing the relationships between simplices, we can uncover important topological features like holes, connectivity, and orientability, laying the groundwork for more advanced concepts in algebraic topology.

Definition of simplicial complexes

• Simplicial complexes are combinatorial objects used to model topological spaces in algebraic topology
• They provide a discrete and combinatorial approach to studying the properties and invariants of topological spaces
• Simplicial complexes are built from simplices, which are the fundamental building blocks

Simplices as building blocks

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• A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions
• A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron
• Higher-dimensional simplices are defined analogously, with an n-simplex being determined by n+1 vertices

Conditions for simplicial complexes

• A simplicial complex is a collection of simplices that satisfies certain conditions to ensure a well-defined topological structure
• Every face of a simplex in the complex must also be in the complex (closure property)
• The intersection of any two simplices in the complex must be either empty or a common face of both simplices (intersection property)

Construction of simplicial complexes

• Simplicial complexes can be constructed in different ways, depending on the context and the desired properties
• Two common approaches are abstract simplicial complexes and geometric realization

Abstract simplicial complexes

• An abstract simplicial complex is defined solely by specifying the vertices and the simplices as subsets of vertices
• It captures the combinatorial structure of the complex without specifying the geometric positions of the vertices
• Abstract simplicial complexes are useful for studying the topological properties and invariants of the complex

Geometric realization

• The geometric realization of a simplicial complex is obtained by assigning specific geometric positions to the vertices in a Euclidean space
• The simplices are then realized as the convex hulls of their vertices, forming a geometric object
• Geometric realization allows for a more concrete visualization and study of the simplicial complex

Simplicial maps

• Simplicial maps are the morphisms between simplicial complexes, preserving the structure and compatibility of the complexes

Definition and properties

• A simplicial map $f: K \to L$ between simplicial complexes $K$ and $L$ is a function that maps vertices of $K$ to vertices of $L$
• The map $f$ must satisfy the condition that if $\{v_0, \ldots, v_n\}$ is a simplex in $K$, then $\{f(v_0), \ldots, f(v_n)\}$ is a simplex in $L$
• Simplicial maps are continuous with respect to the geometric realizations of the complexes

Simplicial approximation theorem

• The simplicial approximation theorem states that any continuous map between the geometric realizations of two simplicial complexes can be approximated by a simplicial map
• This theorem establishes a connection between continuous maps and simplicial maps
• It allows for the study of topological properties using the combinatorial structure of simplicial complexes

Orientation of simplicial complexes

• Orienting a simplicial complex assigns a consistent direction or ordering to the simplices, enabling the study of orientability and related properties

Ordering of vertices

• An ordering of the vertices of a simplex determines its orientation
• For a simplex $\{v_0, \ldots, v_n\}$, an ordering $(v_0, \ldots, v_n)$ or $(v_n, \ldots, v_0)$ defines an orientation
• The orientation changes sign if the ordering is an odd permutation of the vertices

Induced orientation on simplices

• The orientation of a simplex induces an orientation on its faces
• The induced orientation on a face is determined by the order in which the vertices of the face appear in the ordering of the simplex
• Consistent orientation across simplices allows for the study of orientability and related invariants

Simplicial homology

• Simplicial homology is a powerful tool in algebraic topology that captures the "holes" and connectivity of a simplicial complex
• It associates algebraic objects, called homology groups, to a simplicial complex, which provide information about its topological features

Chain complexes and boundary maps

• A chain complex is a sequence of abelian groups (called chain groups) connected by boundary maps
• The n-th chain group $C_n(K)$ is generated by the n-simplices of the simplicial complex $K$
• The boundary map $\partial_n: C_n(K) \to C_{n-1}(K)$ captures the boundary of each n-simplex, with appropriate signs based on orientation

Homology groups of simplicial complexes

• The homology groups $H_n(K)$ of a simplicial complex $K$ are defined as the quotient groups $\ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
• Elements of $H_n(K)$ are equivalence classes of n-cycles (elements in the kernel of $\partial_n$) modulo n-boundaries (elements in the image of $\partial_{n+1}$)
• The rank of $H_n(K)$ counts the number of "n-dimensional holes" in the complex $K$

Invariance under subdivision

• Simplicial homology is invariant under subdivision of the simplicial complex
• Subdividing a simplicial complex (by adding new vertices and simplices) does not change its homology groups
• This property allows for the computation of homology using a convenient subdivision of the complex

Simplicial cohomology

• Simplicial cohomology is the dual notion to simplicial homology, obtained by considering cochains instead of chains
• It associates algebraic objects called cohomology groups to a simplicial complex, capturing additional algebraic structures

Cochain complexes and coboundary maps

• A cochain complex is a sequence of abelian groups (called cochain groups) connected by coboundary maps
• The n-th cochain group $C^n(K)$ is the dual of the n-th chain group, consisting of homomorphisms from $C_n(K)$ to a coefficient group
• The coboundary map $\delta^n: C^n(K) \to C^{n+1}(K)$ is the dual of the boundary map, capturing the coboundary of each n-cochain

Cohomology groups of simplicial complexes

• The cohomology groups $H^n(K)$ of a simplicial complex $K$ are defined as the quotient groups $\ker(\delta^n) / \operatorname{im}(\delta^{n-1})$
• Elements of $H^n(K)$ are equivalence classes of n-cocycles (elements in the kernel of $\delta^n$) modulo n-coboundaries (elements in the image of $\delta^{n-1}$)
• Cohomology groups provide additional algebraic information about the simplicial complex

Cup product structure

• The cup product is an additional algebraic structure on cohomology groups, making them into a graded ring
• It is a bilinear map $\smile: H^p(K) \times H^q(K) \to H^{p+q}(K)$ that combines cohomology classes
• The cup product captures the multiplicative structure of cohomology and provides insights into the ring structure of the complex

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a powerful tool for computing the homology groups of a simplicial complex by decomposing it into simpler pieces
• It relates the homology of a complex to the homology of its subspaces and their intersection

Statement for simplicial complexes

• Let $K$ be a simplicial complex, and let $A$ and $B$ be subcomplexes such that $K = A \cup B$
• The Mayer-Vietoris sequence is a long exact sequence of homology groups: $\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(K) \to H_{n-1}(A \cap B) \to \cdots$
• The maps in the sequence are induced by inclusions and connecting homomorphisms

Computations using Mayer-Vietoris

• The Mayer-Vietoris sequence allows for the computation of homology groups by breaking down a complex into simpler pieces
• By choosing suitable subcomplexes $A$ and $B$, one can often determine the homology of $K$ from the homology of $A$, $B$, and their intersection
• The long exact sequence provides a systematic way to relate the homology groups and perform computations

Simplicial approximation

• Simplicial approximation is a technique for approximating continuous maps between topological spaces by simplicial maps between their simplicial approximations

Continuous maps and simplicial approximation

• Given a continuous map $f: |K| \to |L|$ between the geometric realizations of simplicial complexes $K$ and $L$, a simplicial approximation is a simplicial map $g: K' \to L$ that approximates $f$
• The simplicial complex $K'$ is a subdivision of $K$, obtained by adding new vertices and simplices to refine the structure
• The simplicial map $g$ is chosen to be "close" to $f$ in a suitable sense, often by mapping vertices of $K'$ to nearby vertices of $L$

Applications in algebraic topology

• Simplicial approximation is a fundamental tool in algebraic topology, allowing for the study of continuous maps using simplicial techniques
• It provides a way to transfer problems from the continuous setting to the discrete and combinatorial setting of simplicial complexes
• Simplicial approximation is used in various constructions and proofs, such as the proof of the simplicial approximation theorem and the definition of simplicial homotopy

CW complexes vs simplicial complexes

• CW complexes and simplicial complexes are two important classes of topological spaces used in algebraic topology, each with its own strengths and limitations

Comparison of structures

• Simplicial complexes are built from simplices glued together along their faces, forming a rigid combinatorial structure
• CW complexes are built inductively by attaching cells of increasing dimension, allowing for more flexible constructions
• CW complexes can have cells of different dimensions attached in a less restrictive manner compared to simplicial complexes

Advantages and limitations

• Simplicial complexes have a simple and combinatorial structure, making them well-suited for computational purposes and explicit calculations
• CW complexes provide a more general and flexible framework, allowing for a wider range of topological spaces to be modeled
• However, the increased flexibility of CW complexes can sometimes make computations and explicit descriptions more challenging compared to simplicial complexes

Simplicial sets

• Simplicial sets are a generalization of simplicial complexes that allow for a more flexible and categorical treatment of simplicial objects

Definition and examples

• A simplicial set $X$ is a collection of sets $X_n$ for each non-negative integer $n$, together with face maps $d_i: X_n \to X_{n-1}$ and degeneracy maps $s_i: X_n \to X_{n+1}$ satisfying certain compatibility conditions
• The elements of $X_n$ are called n-simplices, and the face and degeneracy maps encode the relationships between simplices of different dimensions
• Examples of simplicial sets include the standard n-simplex, the nerve of a category, and the singular simplicial set of a topological space

Relationship to simplicial complexes

• Simplicial complexes can be viewed as a special case of simplicial sets, where the simplicial set satisfies additional conditions
• Every simplicial complex gives rise to a simplicial set by taking the set of n-simplices as $X_n$ and defining the face and degeneracy maps accordingly
• However, not every simplicial set arises from a simplicial complex, as simplicial sets allow for more general and flexible structures